$\delta$-$r$-Hyperideals and $\phi$-$\delta$-$r$-Hyperideals of Commutative Krasner Hyperrings

Abstract

In this paper, our purpose is to define the expansion of $r$-hyperideals and extend this concept to $\phi$-$\delta$-$r$-hyperideal. Let $\Re$ be a commutative Krasner hyperring with nonzero identity. Given an expansion $\delta$ of hyperideals, a proper hyperideal $N$ of $\Re$ is called $\delta $-$r$-hyperideal if $a\cdot b\in N$ with $ann(a)=0$ implies that $b\in \delta(N)$, for all $a,b\in\Re$. Therefore, given an expansion $\delta$ of hyperideals and a hyperideal reduction $\phi$, a proper hyperideal $N$ of $\Re$ is called $\phi$-$\delta$-$r$-hyperideal if $a\cdot b\in N-\phi(N)$ with $ann(a)=0$ implies that $b\in\delta(N)$, for all $a,b\in\Re$. We investigate some of their properties and give some examples.